SSP has been studied extensively recently. The works by [1]} and [2]} have achieved a near-optimal regret bound \(\tilde{O}(\sqrt{V_\star B_\star SAK} + B_\star S^2A)\)  when the knowledge on \(B_\star \)  is available to the learner.The upper bound reported in [1]} and [2]} is of order \(B_\star \sqrt{SAK} + B_\star S^2A\) , which is larger than what we report here. This is simply because
in their analysis, they upper bound \(V_\star \)  by \(B_\star \) . We redo their analysis and report their refined dependence on \(V_\star \)  here. Similarly, the lower bound obtained in [5]} is \(B_\star \sqrt{SAK}\)  because in their lower bound construction, they only consider instances where \(V_\star \)  and \(B_\star \)  are of the same order. Hence, the upper bound we obtain here does not violate their lower bound. When such knowledge is unavailable, they design a way to adjust \(B\)  on the fly, and achieve a regret bound of \(\tilde{O}(\sqrt{V_\star B_\star SAK} + B_\star ^3 S^3A)\) .
